Not all economic models conform to linear, quadratic or other standard nonlinear programming formulations. Rather, such models require the solution of highly non-linear equations systems using nonstandard and innovative, iterative algorithms that exploit the special features of those equations. Numerical solutions of models using iterative techniques have been a goal, though poorly practiced within the field of transportation and land use modeling. Meanwhile, iterative numerical methods are gaining broader applicability within economics to solve a variety of problems. For example, a survey and exposition of such methods and problems is described in Numerical Methods in Economics. See Kenneth L. Judd, Numerical Methods in Economics (MIT Press 1998).
A brief overview of relevant art is helpful in order to understand the deficiencies in the known methodologies. Lowry's model in 1964 was the first to recognize the importance of building a computable model of a metropolitan area. See Ira S. Lowry, A Model of Metropolis (RM-4035-RC, RAND Corp., 1964). Lowry's model was limited by the available data and lack of any prior theory, since urban economics had barely emerged in 1964. Thus, Lowry used crude gravity models with ad-hoc equilibration of land use. This modeling style was durably influential on a subsequent line of developments commonly known as “Lowry-type models”, e.g., DRAM-EMPAL, which models lacked economic content (See S. H. Putman, Integrated Urban Models (Pion Press 1983)), even after data became available and existing theories improved. Following Lowry, however, there were two important benchmark contributions by economists.
First, the NBER model of housing markets made strides in computability, by introducing better microeconomic content and emphasizing policy applicability. See Gregory K. Ingram, John F. Kain, and J. R. Ginn, The Detroit Prototype of the NBER Urban Simulation Model (National Bureau of Economic Research 1972). This work inspired extensions to models that culminated in the Anas and Arnott model of housing markets. See Alex Anas and Richard J. Arnott, Dynamic Housing Market Equilibrium with Taste Heterogeneity, Idiosyncratic Perfect Foresight and Stock Conversions, Journal of Housing Economics, 1, 1, 2-32 (1991); Alex Anas and Richard J. Arnott, Taxes and Allowances in a Dynamic Equilibrium Model of Urban Housing Market with a Size-Quality Hierarchy, 27 Regional Science and Urban Economics 547, 547-580 (1997). Second, the general equilibrium model of metropolitan structure developed by Mills and extended by Hartwick and Hartwick, and Kim, utilized a linear programming based fixed-coefficient technology, but included a sophisticated treatment of traffic congestion on a grid geography with endogenous road capacities. See Edwin S. Mills, Markets and Efficient Resource Allocation in Urban Areas, 74 Swedish Journal of Economics 100, 100-113 (1972); Philipp G. Hartwick and John M. Hartwick, Efficient Resource Allocation in a Multinucleated City with Intermediate Goods, 88 Quarterly Journal of Economics 340, 340-352 (1974); T-J. Kim, Alternative Transportation Modes in a Land Use Model: A General Equilibrium Approach, Journal of Urban Economics, 6, 2, 197-215 (1979). A dynamic version was later developed by Moore in his doctoral dissertation. See James E. Moore and Lyna Wiggins, A Dynamic Mills Heritage Model with Replaceable Capital, 68 Papers of the Regional Science Association 23, 23-41 (1990). The linear programming structure used in the aforementioned models, although fully consistent with economic theory and solvable using standard methods and the earlier partial equilibrium model of Herbert and Stevens, have limitations that affect empirical relevance, computability and ease of calibration. See John D. Herbert and Benjamin H. Stevens, A Model for the Distribution of Residential Activity in Urban Areas, 2 Journal of Regional Science 21, 21-36 (1960). These limitations were overcome by the use of discrete choice models which are better suited to treating heterogeneity as explained by Anas, and supported by Harris. See Alex Anas, Residential Location Markets and Urban Transportation: Economic Theory, Econometrics and Policy Analysis with Discrete Choice Models (Academic Press 1982); Britton Harris, Urban Simulation Models in Regional Science, Journal of Regional Science, 25, 4, 545-567 (1985). In the 1990s Anas-Kim, Anas-Xu and more recently Anas-Rhee extended the prior body of work to general equilibrium formulations by using discrete choice to model the joint choice of workplace, residence and housing type while unifying it with the Dixit and Stiglitz representation of utility and production functions to generate budget-constrained discretionary trips for consumers and inter-industry linkages for producers. See Alex Anas and Ikki Kim, General Equilibrium Models of Polycentric Urban Land Use with Endogenous Congestion and Job Agglomeration, 40 Journal of Urban Economics 232, 232-256 (1996); Alex Anas and Rong Xu, Congestion, Land Use and Job Dispersion: A General Equilibrium Analysis, Journal of Urban Economics, 45, 3, 451-473 (1999); Alex Anas and Hyok-Joo Rhee, Curbing Urban Sprawl with Congestion Tolls and Urban Boundaries, 36 Regional Science and Urban Economics, 510, 510-541 (2006); Avinash Dixit and Joseph Stiglitz, Monopolistic Competition and Optimum Product Diversity, American Economic Review, 67(3), 297-308 (1977).
As the aforementioned models were developing, the modeling of equilibrium traffic on congested highway networks by transportation scientists, evolved largely separate from urban economics. Florian and Nguyen provided one of the earliest, while Bar-Gera provided one of the most recent, algorithms that find traffic equilibria on arbitrarily configured highway networks with fixed capacities, making operational the mathematical programming formulation of static traffic flow by Beckmann, McGuire and Winsten. See Michael Florian and S. Nguyen, An Application and Validation of Equilibrium Trip Assignment Models, 10 Transportation Science 374, 374-390 (1976); Hillel Bar-Gera, Origin-Based Algorithms for the Travel Assignment Problem, 36 Transportation Science 398, 398-417 (2002); Martin Beckmann, C. B. McGuire, and C. B. Winsten, Studies in the Economics of Transportation, Cowles Foundation for Research in Economics, Yale University (1958). However, being focused on transportation, these models take the land use distribution and, often, the distribution of zone-to-zone trips as being fixed. This contrasts with urban economic theories where the interdependence of transportation and land use has been center-stage from the beginnings of the field of study.
It has been recognized since the 1960s that a consistent operational metropolitan model, should integrate a model of congested traffic network equilibrium with a model of land use. This has led to a variety of efforts to devise integrated transportation and land use models in both academia and planning practice without proper grounding in economics. In one line of such model integration, a rigorous, well-documented but partial form of model integration is sometimes attempted as an extension of travel demand analysis in order to improve travel forecasts. In such efforts, the origin-zone to destination-zone trip matrices are not fixed, as is common in conventional travel analysis, but respond in some cost-sensitive way to equilibrium travel costs on the networks. These “integrated models of origin-destination, mode and route choice” or “combined models” date back to Evans. See Suzanne P. Evans, Derivation and Analysis of Some Models for Combining Trip Distribution and Assignment, 10 Transportation Research 37, 37-57 (1976). Only the trip generation, distribution, mode choice and network equilibrium steps of travel forecasting are combined or treated with feedbacks, but land use remains implicitly fixed. The state-of-the-practice in these combined models is well-documented in a recent presentation by Boyce and Florian, including large scale “computational experiments for various integrated models of variable demand and auto route choice for a single, aggregated class of travelers plus trucks.” See David E. Boyce and Michael Florian, Workshop on Traffic Assignment with Equilibrium Methods, Subcommittee on Network Equilibrium Modeling, Transportation Network Modeling Committee, Transportation Research Board, Jan. 9, 2005. Although combined models are a step forward from the traditional four-step model of sequential travel forecasting, none of these integration experiments include the variable travel demands made consistent with a land use model, nor is a fully economic land use or regional economy model made to respond to these travel demands. According to Boyce and Florian, although the combined models used in planning practice show convergence to equilibrium in some large scale applications, they may not always converge well or to a high level of accuracy, but scholars (See, e.g., Bar-Gera and Boyce) are continuing to develop faster algorithms.
In a second line of applications in metropolitan planning by consultants and practitioners, successful transportation and land use model integration has been a goal. Models that have reportedly attempted this include MEPLAN, or the closely related TRANUS model of De la Barra. See Tomas De la Barra, Integrated Land Use and Transport Modeling (Cambridge University Press 1989). These are not fully documented in scientific journals or in other publicly available forms, and cannot therefore be understood or evaluated completely. The MUSSA model of Martinez is better documented in scientific journals. See Francisco Martinez, The bid-choice land use model: an integrated economic framework, 15 Environment and Planning A 871, 871-885 (1992); Francisco Martinez, MUSSA: A Land Use Model for Santiago City, Transportation Research Record 1552: Transportation Planning and Land Use at State, Regional and Local Levels, 126-134 (1996). The popular do-it-yourself modeling template of Wadell, UrbanSim, is open source and it is easy to verify that prices are not market clearing and that it does not conform to economic theory. See Paul Waddell, An Urban Simulation Model for Integrated Policy Analysis and Planning: Residential Location and Housing Market Components of UrbanSim, Proceedings of the 8th World Conference on Transport Research, Antwerp, Belgium July 12-17 (1998). Model integration in practice is highly demanding of technical personnel.
In the past, antecedents of the instant invention that were not as general were used to evaluate the effects of planned transportation investments on real estate prices. See Alex Anas and Liang-Shyong Duann, Dynamic Forecasting of Travel Demand, Residential Location and Land Development: Policy Simulations with the Chicago Area Transportation/Land Use Analysis System, 56 Papers of the Regional Science Association 38, 38-58 (1985). After the investments took place and price changes were measured, other models, McDonald and Osuji (1995) and McMillen and McDonald (2004), claimed that the forecasts were accurate. See John F. McDonald and Clifford I. Osuji, The Effects of Anticipated Transportation Improvement on Residential Land Value, 5 Regional Science and Urban Economics 261, 261-278 (1995); Daniel McMillen and John F. McDonald, Reaction of House Prices to a New Rapid Transit Line: Chicago's Midway Line, 1983-1999, Real Estate Economics, 32(3), 462-486 (2004). In another model by Anas and Arnott (1997), a dynamic housing market model was used to compare conflicting aspects of taxation and of subsidies on the Chicago housing market. See Alex Anas and Richard J. Arnott, Taxes and Allowances in a Dynamic Equilibrium Model of Urban Housing Market with a Size-Quality Hierarchy, 27 Regional Science and Urban Economics 547, 547-580 (1997). While the same model was used in four MSAs to compare the welfare effects of demand-side and supply-side housing subsidization policies on the welfare of consumer groups. See Alex Anas and Richard J. Arnott, Development and Testing of the Chicago Prototype Housing Market Model, Journal of Housing Research, 4 (1), 73-130 (1993). Recently, another model adopted the stationary version of the same dynamic model to examine the causes of homelessness in California and the effects of various policies on the homeless. See Erin. T. Mansur, John M. Quigley, Steven Raphael and Eugene Smolensky, Examining Policies to Reduce Homelessness Using a General Equilibrium Model of the Housing Market, 52 Journal of Urban Economics 316, 316-340 (2002).
Thus, there is a long-felt need for a method of modeling the interrelations between regional economic, land use and transportation needs for a metropolitan area which is capable of responding to varying travel demands. There also is a long-felt need for a method of modeling the interrelations between regional economic, land use and transportation needs for a metropolitan area in order to predict the effects of changes to a variety of characteristics of the metropolitan area.